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Theory of Multisource Crosstalk Reduction by Phase-Encoded Statics

Theory of Multisource Crosstalk Reduction by Phase-Encoded Statics. G. Schuster, X. Wang, Y. Huang, C. Boonyasiriwat King Abdullah University Science & Technology. Outline. Seismic Experiment:. L m = d. 1. 1. L m = d. 2. 2. L m = d. N. N.

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Theory of Multisource Crosstalk Reduction by Phase-Encoded Statics

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  1. Theory of Multisource Crosstalk Reduction by Phase-Encoded Statics G. Schuster, X. Wang, Y. Huang, C. Boonyasiriwat King Abdullah University Science & Technology

  2. Outline Seismic Experiment: Lm = d 1 1 Lm = d 2 2 Lm = d . . N N . 2. Standard vs Phase Encoded Least Squares Soln. L d 3. Theory Noise Reduction ]m = [N d + N d ] vs [ N L + N L 1 1 m = L d 2 2 2 2 1 1 1 1 2 2 4. Summmary and Road Ahead

  3. Gulf of Mexico Seismic Survey 4 d Predicted data Observed data 1 Goal: Solve overdetermined System of equations for m Time (s) Lm = d 1 1 Problem: Expensive, one migration/shot gather 0 Lm = d 6 X (km) 2 2 Solution:Supergather migration Lm = d . . N N . m

  4. Brief History Multisource Phase Encoded Imaging Migration Romero, Ghiglia, Ober, & Morton, Geophysics, (2000) Waveform Inversion and Least Squares Migration Krebs, Anderson, Hinkley, Neelamani, Lee, Baumstein, Lacasse, SEG, (2009) Virieux and Operto, EAGE, (2009) Dai and Schuster, SEG, (2009) Biondi et al., SEG, (2009)

  5. Outline Seismic Experiment: Lm = d 1 1 Lm = d 2 2 Lm = d . . N N . 2. Standard vs Phase Encoded Least Squares Soln. L d 3. Theory Noise Reduction ]m = [N d + N d ] vs [ N L + N L 1 1 m = L d 2 2 2 2 1 1 1 1 2 2 4. Summmary and Road Ahead

  6. Conventional Least Squares Solution: L= & d = L d 1 1 L d Given: Lm=d 2 2 In general, huge dimension matrix Find: m s.t. min||Lm-d|| 2 Solution: m = [L L] L d -1 T T or if L is too big m = m – aL (Lm - d) (k+1) (k) (k) T (k) = m – aL (L m - d ) (k) (k) [ ] T + L (L m - d ) T 1 1 1 2 2 2 Note: subscripts agree Problem: L is too big for IO bound hardware

  7. Conventional Least Squares Solution: L= & d = L d 1 1 L d Given: Lm=d 2 2 In general, huge dimension matrix Find: m s.t. min||Lm-d|| 2 Solution: m = [L L] L d -1 T T m = m – aL (Lm - d) (k+1) (k) (k) T (k) = m – aL (L m - d ) (k) (k) [ ] T + L (L m - d ) T 1 1 1 2 2 2 Problem: Expensive, FD solve/CSG Solution: Blend+encode Data Problem: L is too big for IO bound hardware

  8. Blending+Phase Encoding Blending Blending Phase Phase 1 d d d Lm= Lm= Lm= 1 1 3 2 2 3 Encoded supergather O(1/S) cost! Encoding Matrix d = N d + N d + N d in w domain 1 1 2 3 2 3 Encoded supergather modeler L = NL + N L + NL m [ ]m 1 2 3 1 2 3 e iwt

  9. Blended Phase-Encoded Least Squares Solution L =&d = N d + N d N L + N L 1 2 1 1 1 2 2 2 Given: Lm=d In general, SMALL dimension matrix Find: m s.t. min||Lm-d|| 2 Solution: m = [LL] Ld -1 T T or if L is too big m = m – aL (Lm - d) T (k+1) (k+1) (k) (k) (k) (k) (k) = m – aL (L m - d ) (k) (k) (k) (k) [ (k) (k) ] (k) (k) T + L (L m - d ) T (k) Iterations are proxy For ensemble averaging 1 1 1 2 2 2 + Crosstalk * * (k) + L N N (L m - d ) (k) (k) (k) (k) (k) (k) T L N N (L m - d ) (k) T + (k) (k) (k) (k) 2 1 2 1 1 1 2 2 2 1

  10. Outline Seismic Experiment: Lm = d 1 1 Lm = d 2 2 Lm = d . . N N . 2. Standard vs Phase Encoded Least Squares Soln. L d 3. Theory Noise Reduction ]m = [N d + N d ] vs [ N L + N L 1 1 m = L d 2 2 2 2 1 1 1 1 2 2 4. Summary

  11. Ensemble Average of Crosstalk Term With Random Time Shifts e -iwt 2 + Crosstalk: Noise = e e e iw(t - t ) iwt <Noise>: < > -iwt < > = < > 1 1 2 2 Gaussian PDF -2 e iw(t - t ) -.25 s (t - t ) e = 1 2 1 2 2 2 Crosstalk term decreases with increasing w and s -s w e ~ dtdt 1 2 * * * L N N (L m - d ) N N T L N N (L m - d ) T 1 1 2 1 2 1 1 1 2 2 2 2 1

  12. Crosstalk Prediction Formula ~ X = O( ) 2 2 e -s w Pt. Scatt. Stand. Mig. Pt. Scatt.. Mig. of Supergathers. Depth s = .01 s X Offset Pt. Scatt.. Mig. of Supergathers. Pt. Scatt.. Mig. of Supergathers. s = .05 s s = .1 s .01 1.0 s L (L m - d ) + L (L m - d ) T T 2 1 1 2 1 2

  13. sgn(t ) Ensemble Average of Crosstalk Term With Random Polarity sgn(t) + Crosstalk: Noise < > = < > sgn(t ) sgn(t ) <Noise>: = 0 1 2 Conclusion: Random polarity better than random time shifts Further Analysis: Variance of the crosstalk noise says that random polarity & random time shifts can be almost twice better than polarity alone ( ) 2 < > * * * * L N N (L m - d ) N N N N T L N N (L m - d ) T 1 1 1 2 1 2 1 1 1 2 2 2 2 2 1

  14. Dt < +/- < +/-, Dt < +/-, Dt, Dx Key Theory+Num. Results for 320 CSG Supergather (Xin Wang, Yunsong Huang) Time Statics Polarity Polarity+Time Statics Polarity+Time Statics+Location Statics b) Time static σ = 0.1 s c) Noise a) Standard migration (320 CSG) a) Polarity b) Noise 0.4 0.4 0.18 0.4 0.4 0.4 Z (km) Z (km) Z (km) Z (km) Z (km) SNR 1.48 1.48 1.48 1.48 1.48 0 Time static σ (s) X (km) X (km) X (km) X (km) X (km) 6.75 6.75 6.75 6.75 6.75 0.1 0.01 0 0 0 0 0 e) Noise f) SNR d) Source polarity & static Polarity and time static polarity Time static

  15. Key Results Theory of Multisource Imaging of Encoded Supergathers(Xin Wang) Sig/Noise = GI < GIN b) Image of 5 stacks a) Image of 1 stack 0 # geophones/supergather # subsupergatherss Z (km) # iterations 0 0 Z (km) Z (km) 1.5 X (km) X (km) 6.75 6.7 X (km) 0 0 0 6.7 d) SNR vs Iterations c) Image of 50 stacks 1.5 1.5 10 Prediction Bulk shift SNR Observed 1 115 1 Iteration Number

  16. ~ ~ 1 GI G GS [s(t) +n(t) ] S S Standard Migration SNR Zero-mean white noise Assume: d(t) = Standard Migration SNR Neglect geometric spreading GS Cost ~ O(S) # CSGs # geophones/CSG + + + Iterative Multisrc. Mig. SNR Cost ~ O(I) Standard Migration SNR SNR= # iterations migrate stack migrate iterate . SNR= . . SNR=

  17. Outline Seismic Experiment: Lm = d 1 1 Lm = d 2 2 Lm = d . . N N . 2. Standard vs Phase Encoded Least Squares Soln. L d 3. Theory Noise Reduction ]m = [N d + N d ] vs [ N L + N L 1 1 m = L d 2 2 2 2 1 1 1 1 2 2 4. Summary

  18. 3. Summary ]m = [N d + N d ] N L + N L [ L d 2 2 2 2 1 1 1 1 1 1 m = L d vs O( ) 2 2 Dt < +/- < +/-, Dt < +/-, Dt, Dx GI GS 2 2 e -s w 1. ~ < > Time Statics Polarity Polarity+Time Statics Polarity+Time Statics+Location Statics 2. SNR: VS 4. Passive Seismic Interferometry = Multisrc Imaging L (L m - d ) + L (L m - d ) T T 2 1 1 2 1 2

  19. Summary L d 1 1 m = ]m = [N d + N d ] N L + N L [ L d 2 2 2 2 1 1 1 1 2 2 Stnd. MigMultsrc. LSM IO 1 1/320 1 <1/10 Cost ~ Less 1 1 SNR~ Resolution dx 1 1 Cost vs Quality

  20. The SNR of MLSM image grows as the square root of the number of iterations. 7 GI SNR = SNR 0 300 1 Number of Iterations

  21. Dt < +/- < +/-, Dt < +/-, Dt, Dx Key Theory+Num. Results for 320 CSG Supergather (Xin Wang, Yunsong Huang) Time Statics Polarity Polarity+Time Statics Polarity+Time Statics+Location Statics b) Time static σ = 0.1 s c) Noise a) Standard migration (320 CSG) a) Polarity b) Noise 0.4 0.4 0.18 0.4 0.4 0.4 Z (km) Z (km) Z (km) Z (km) Z (km) SNR 1.48 1.48 1.48 1.48 1.48 0 Time static σ (s) X (km) X (km) X (km) X (km) X (km) 6.75 6.75 6.75 6.75 6.75 0.1 0.01 0 0 0 0 0 e) Noise f) SNR d) Source polarity & static Polarity and time static polarity Time static

  22. Key Results Theory of Multisource Imaging of Encoded Supergathers(Xin Wang) Sig/Noise = GI < GIN b) Image of 5 stacks a) Image of 1 stack 0 # geophones/supergather # subsupergatherss Z (km) # iterations 0 0 Z (km) Z (km) 1.5 X (km) X (km) 6.75 6.7 X (km) 0 0 0 6.7 d) SNR vs Iterations c) Image of 50 stacks 1.5 1.5 10 Prediction SNR Observed 1 115 1 Iteration Number

  23. Key Results Theory of Multisource Imaging of Encoded Supergathers(Boonyasiriwat) Sig/Noise = GI < GIN # geophones/supergather # subsupergatherss # iterations Dynamic Polarity Tomogram (1089 CSGs/supergather) 3.5 km 1/300 1/1000 Dynamic QMC Tomogram (99 CSGs/supergather)

  24. Ld +L d 1 2 2 1 = L d +L d + =[L +L ](d + d ) 1 2 1 2 1 1 2 2 d +d =[L +L ]m 1 2 1 2 mmig=LTd Multisource Migration: Multisource Phase Encoded Imaging { { L d Forward Model: T T T T T T m = m + (k+1) (k) Crosstalk noise Standard migration

  25. Dt < +/- < +/-, Dt < +/-, Dt, Dx Time Statics Polarity Polarity+Time Statics Polarity+Time Statics+Location Statics Dt < +/- < +/-, Dt < +/-, Dt, Dx Relative Merits of 4 Encoding Strategies Time Statics Polarity Polarity+Time Statics Polarity+Time Statics+Location Statics d d d Lm= Lm= Lm= 1 1 3 2 2 3 Supergather #1 Supergather #2 Supergather #3 Supergather #4 supergather

  26. Ld +L d 1 2 2 1 = L d +L d + =[L +L ](d + d ) 1 2 1 2 1 1 2 2 d +d =[L +L ]m 1 2 1 2 mmig=LTd Multisource Migration: Phase Encoded Multisource Migration { { d L Forward Model: mmig T T mmig T T T T + + Crosstalk noise mmig T T T T = L d +L d + Ld +Ld 1 2 2 1 1 1 2 2 mmig = L d +L d 1 1 2 2 Standard migration

  27. Ld +L d 1 2 2 1 = L d +L d + =[L +L ](d + d ) 1 2 1 2 1 1 2 2 d +d =[L +L ]m 1 2 1 2 mmig=LTd Multisource Migration: Phase Encoded MultisrceLeast Squares Migration { { L d Forward Model: mmig T T T T T T m = m + (k+1) (k) Crosstalk noise Standard migration

  28. Outline Seismic Experiment: Lm = d 1 1 Lm = d 2 2 Lm = d . . N N . 2. Standard vs Phase Encoded Least Squares Soln. L d 3. Theory Noise Reduction ]m = [N d + N d ] vs [ N L + N L 1 1 m = L d 2 2 2 2 1 1 1 1 2 2 4. Numerical Tests

  29. RTM & FWI Problem & Possible Soln. Problem: RTM & FWI computationally costly Solution: Multisource LSM & FWI Preconditioning speeds up by factor 2-3 LSM reduces crosstalk

  30. d +d =[L +L ]m 1 2 1 2 mmig=LTd Multisource Migration: Multisource Least Squares Migration { { L d Forward Model: Phase encoding Kirchhoff kernel Standard migration Crosstalk term

  31. Multisource Least Squares Migration Crosstalk term

  32. Conventional Least Squares Solution: L= & d = L d 1 1 L d Given: Lm=d 2 2 In general, huge dimension matrix Find: m s.t. min||Lm-d|| 2 Solution: m = [L L] L d -1 T T or if L is too big m = m – aL (Lm - d) (k+1) (k) (k) T (k) = m – aL (L m - d ) (k) [ ] T + L (L m - d ) T 1 1 1 2 2 2 Note: subscripts agree Problem: L is too big for IO bound hardware

  33. Key Results Theory of Multisource Imaging of Encoded Supergathers(Boonyasiriwat) Sig/Noise = GI < GIN # geophones/supergather # subsupergatherss # iterations Dynamic Polarity Tomogram (1089 CSGs/supergather) 3.5 km 1/300 1/1000 Dynamic QMC Tomogram (99 CSGs/supergather)

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